Brownian motion, Ito's lemma, an÷♥d the Black-Scholes formula (Par≥₩φt II)
發布時(shí)間(jiān)：2017-03-22 | ↓₩ 來(lái)源:
作(zuò)者：石川
1 Recap
In the first article of≥∏ Ω this series, we explained the properties ↑→↕™of the Brownian motion as well as ≥↑why it is appropriate to use the geomet≤÷Ωric Brownian motion to model s✘•tock price movement. In ad≤λdition, we gave the b¶∑asic form of Ito's lemma, whi₹₹↑≤ch paved the way for stochastic calcu£ lus and derivative pricing. In today's article, we w¥Ω©ill begin with the general form ★&•∞of Ito's lemma and u¥↕↕se it to solve the geometric φ≥Brownian motion, and derive the B•φ→ lack-Scholes differential equat♦∏ions and the Black-Scholes formula ×↔σfor European option pricing (this is al₽§so referred to as the §₹Black-Scholes-Merton formula). To der✘<♣φive the BS formula, we will≤↕♦ focus on a crucial ide φ>a in derivative pricing, the risk-neu₽₩↓γtral valuation. We wiΩγ€σll also put some effort to explai≥σπn N(d_1) and N(d_2) which are two key £∑™¶elements in the BS formu₹←×₽la. An understanding of them is e☆∑®ssential to mastering theγ♥ BS formula.
Before we step into te¥®©>chnical details, let÷±₹'s have some fun first. The fig≠≠δure below shows Black, Merton, and÷ Scholes. Scholes and Merton received the Nobel Me∏¶morial Prize in Economic Sσαciences in 1997 for their outst <"anding contributions in∞€♦♦ derivative pricing. The ₩©→≈reason why Black was not awar≈♥&ded is that he passed aw★σay in 1995 and the Nobel prize i ®¶¥s not posthumously awarded to ♣♠γscholars who died for more than 6∞β months.
2 Ito's lemma
A Brownian motion with drift and diffu<¶♣sion satisfies the following stoc∞$γ∑hastic differential equati™ on (SDE), where μ and σ are so me constants
More generally, the drift and dif"±¶£fusion coefficients can be function≈♣ s of X(t) and t, rather than som♥÷←←e constants. Let a(X(t),t$π ) and b(X(t),t) be the↑™← drift and diffusion coeffic₹±∏ients (in the example above, we the $∏refore have a(X(t),t) = μ and b(X(t),t)✔>" = σ). We define a stochastic procesα$§s that satisfies the following SDE ∑<δ≤the Ito drift-diffusion process:
Let f(X(t), t) be a continuous>≥ ¶ and smooth function of X(∏σ→φt) and t. With Ito's lem¶&←↕ma, we derive:
Substitute dX with a(X(t)✔↑<☆,t)dt + b(X(t),t)dB in the equation ↔∑πabove and ignore the higher δα₹order terms of dt. We finally ₹βαderive the general form of Ito's lemma:
From the SDE of f, it is easy to s®₹→☆ee that as a function of X and t, f its¶ ®elf is also a Ito drift-diffusio&←n process. More importantly, comparing the ex₽ε₽εpressions of dX and df, We see 'ε÷that both X and f are ✔∏affected by the same underlying s♥ource of uncertainty,©"↓Ω dB. In other words, the randomness po↓"ssessed by the stocha✔♥&stic processes X and&♥ f comes from the same Brφ×ownian motion. This proves to be ve↓±ry important in the derivationβ≤ of the BS formula.
We next apply Ito's lemma to solδ☆€ve the geometric Brownλ<↕ian motion.
3 Geometric Brownian motion
We model the price of some stock usin<'αg the geometric Brownian m"λ otion. Let S denote ₹¥the price of the stock, επ£then it satisfies the foll±✘©owing SDE
where μ is the annual rate of return ✔¶and σ is the standard deviatioβ±®n of that return. S is ther→₩↕efore a Ito process where a = μS₹∑★$ and b = σS. To solve S, let f = l∞≈nS and apply Ito's lemma to df to deri✘&ve the SDE of lnS
The result shows that ♦λlnS is a Brownian motion with•≤ drift rate of μ – 0.×$♠ 5σ^2 and diffusion rate of σ. Accord¶₩ing to the property of the Browni∑>€•an motion, within any interval [±$0, T], lnS (T) satisfies the follow£↕ing normal distribution:
A variable has a lognormal distribution≠φ♠ if the natural logarithm of t$$✔he variable is normally distrλ♦ibuted. Therefore, when we model stocδ ¥✘k price using a geomeπ¶€γtric Brownian motion, the stock price satisfies a lognor$δ←♥mal distribution. Taking the integral of the SDE ׶of lnS and then taking tβ×he exponential of the result, it i¥±←s easy to derive how S var→÷£ies with T:
This may seem counter-intuitive at th↔↔←e first glance. We assumed that the annual rate≤∑α of return of the stock is μβ§✘±. However, in the formula ™→above, if we neglect the ra©∞♦ndomness from B(T) and onl±y focus on the first part, it says ✘ that the stock price grows★↓ at a rate of μ – 0.5σ^2 rather than→α μ. What does this mean? What does th ≥e rate μ – 0.5σ^2 stand for? The right answer is that μ –'$₩ 0.5σ^2 is the continuously compounded rate ©∑®of return per annum. This can be explained as f Ωollows. Let x be the contin☆¥uously compounded raε÷↔≈te per annum, and then S(T) follows S(÷'T) = S(0)e^(xT) or x = (1/T←σ&)×(lnS(T) - lnS(0)). Sin★♥±ce S is lognormally distributed, we €∞know that lnS(T) – lnS(0∑&>) follows a normal distribution ←¥with mean (μ – 0.5σ^2)T and $Ωγvariance (σ^2)T. We can then solve₽→≈ for the x and it satisfies the folφ≤lowing normal distribu÷≈ ≠tion:
While μ is the expected annual return, β μ – 0.5σ^2 is the continuo±×usly compounded rate of re₹∏☆turn where it considers the va ✘riation σ. There difference is analog tλ©©δo the difference between the∑® arithmetic mean and the geometric mσ÷ean of return time series.
Consider the followin&≤ g example. Suppose that i>±n the past 5 years a stoc§≥'k has returns of 15%, 20%, 30%, -20% a♥←®nd 25%. Their arithmetic mean ±§↑÷is 14% and it means that the expected✔€ annual return μ is 14%. What ×☆φφis the continuously compounded rate ofα↕ return? If we spent $ε£100 to purchase this stock 5 years a→☆♠go and hold it since then, todayφ≤α☆ our payback will be $100 × 1.15 × 1.2÷£"0 × 1.30 × 0.80 × 1.25 = $€α179.4. This is equivalent to a 12.4% c§ ↕ontinuously compounded r ate of return. It is lower than 14% du$''¥e to the volatility of♥• ∑ this return series.
4 Black-Scholes differential equatio♠•n
In this section, we introduce the B₹♦★lack-Scholes differen£₩tial equation. Note that→∏∏← the term 'stochastic' does not apφ∏™pear here, which means t€♦hat the BS equation h $as no randomness, and this iδ"s great because no one likes u→≥♥ncertainty. You will know wh→'ε±y it is the case after reading thi→♥©s section. The assumptions used to derive the BS d®∏€ifferential equation are as fo"♦σllows (Hull 2011):
1. European options are ♠ ≠↑considered where they♣" can be exercised only at maturity.φ♣
2. The stock price foλ∏≤llows the geometric Brownian ¶↕motion.
3. The short selling of securities w÷ ✔ith full use of proceed♦←'s is permitted.
4. There are no transa σ≥ctions costs or taxes. All securitie∑ s are perfectly divisble→.
5. There are no dividends during÷¥δ the life of the der≤₩ivative.
6. During the life of tε∑εhe option, the standard deviati∞♠→∑on σ of the return of the ♠£>™underlying security is k£φ♠nown and constant.
7. There are no riskless arbitr≤ age opportunities.
8. Security trading is cont$φ₽↓inuous.
9. The risk-free rate of inter♦×↑est, r, is constant and the sa€÷me for all maturitiesε₽.
It is true that some assumptions are no$πt realistic. However, they ar☆≈e generally accepted fo¥±r the purpose of pricinβg the options. Since the BS model wasΩ first proposed, academia has madδ♣e great progress in derivative pricin→€≤Ωg. A number of improved models hav♠₹α&e been developed to $δ&¶correct various assumptions 'in the BS model. We next use the Europ¶¥ean call option as an exa¶≥$<mple to introduce the BS differ→∑$¶ential equation.
Let C be the price of the call opt≈≈λion. It is a function of S -↕♠- the price of the underlying stocφ≥↕k -- and time t. Applying Ito'sεφ₹• lemma to C(S, t) gives<↕₽
Let's consider how S ו€δand C change within an infin≠β↔εitesimal time interval Δt. To do so, consider theΩ↓€♥ discrete versions of dS and dC:
We mentioned in section 2 of this€♥ φ article that the function f of a ♦≤™≠Ito process X is also a Ito process,βγ and more importantly, the uncertaint←♥"y of f and X comes from the same underl♣★ying Brownian motion. This means that the unc∏$≤ ertainty in ΔC and ΔγS comes from the same ΔB. It follows that a portfol≈>≤Ωio of the stock and this call option ca•♣≈n be constructed so that thi&™s Brownian motion is eliminated. This portfolio is:
The holder of this portfolio isβ£' short one derivative and long a¥↔n amount ∂C/∂S of shares of the stock. ±λ¥₽It is easy to see that with≥©♠ such weights, the Brownian mo©£tion ΔB is perfectly eliminated in↕≈$ this portfolio. This techn>±>ique of creating portfolio is c₩↕Ω<alled Delta hedging. Let P be the value of the portfoli¶¶♦αo, then P's change within the in→'>terval Δt is
Since ΔB does not appear in the €¥♣βexpression of ΔP, it is only a funct∏¥₩ion of time. By shorting onΩ₩e option and longing ∂C/∂S shares of st ★₹♠ock, we have cancelled&λ≥ the risk within the time↕σ★♥ interval Δt and constructe☆♦"d a riskless portfolio. In a arbitrage-free market₹₹¥π, this portfolio must earn the risk-f↑ree return of r, and therefore ΔP = rPΔt. ↔→Substituting ΔP and P = -C + (€∑₽∂C/∂S)S into this equation g↔εives
This is the famous Black-Schπ♠₩☆oles differential equation. Since we removed the rando>©βmness by delta hedging, there is no stochastic term in th↔§≈>is equation, and therefore it is a regulaβ✔₩↑r (partial) differential equation,≠✔¶ rather than a stochastic differ☆φ" ential equation. Solving this differential equation"∞ requires the correct boundary cπ©↓onditions. For European call op≥→αtion, its boundary condition is that a ☆≥t t = T (when the option matures), δφC must satisfy C = max(S(T) - K, 0♠®$δ), where K is the stri♠←€ke price.
5 Risk-neutral valuation
It is true that we ca∏≥©n derive the price of t∏γ≥★he call option by so∏™lving the differential equation w≥÷ith appropriate boundaryΩ∏αλ conditions. However, in derivative pr¶¥♠icing, there is another v×¥§∑ery useful method that could not be emp©∏hasized too much. It i<₩s called risk-neutral v±∑aluation.
You may get confused and π₩ ask if that is the case why wε✘e put a lot of effort to derive thβ∑e BS differential equati '∏♥on. However, risk-neutral valuation comes from ±✘®↔one key property of the BS different≈≤ial equation: "The BS differential equation does no'≠≠×t involve any variables★π'≠ that are affected by the risk prefere≥★ ¶nces of investors. The÷™★∑ variables that do appear in th≈§λe equation are the current s₹ αtock price, time, stock price volati¶$ε↑lity, and the risk-free rat≤δe of interest. All are independe≤>∑nt of risk preferences."
We see from the BS differential ₹φequation that the exε≤☆®pectation return of the stock, μ, does®≈ not appear in it. Appar &↓ently, μ does depend on risk prefer₹≠₹ences of investors. The higher t☆§he level of risk aversion by investo∑>πrs, the higher μ will be for an₩↕y given stock. It is ↑₩fortunate that when we use delta¥♥<≥ hedging to remove the Brownian motion,> μ also happens to drop out in the derα↑✘∞ivation. This is a happy acciden→←t.
Because the BS differenti≠"αal equation is independent of ris✔≥☆∞k preferences, it meanγλ♠s it cannot impact its solut↑♣♥ion. Any set of risk preferenc↓>©es can be used when evaluating C. In φ"particular, the very simple assumption>→£ that all investors are★€Ω risk neutral can be made®₹✘ .
To price any derivativ ∑♦e, we must know two things:
1. Its expected price at its maturit'σ<←y. Since the price of a derivative <≠♥is a function of the ✘£price of the underlying asset, i≈t is therefore dependent of the exp↕εected return of the under✘&lying asset, μ.
2. After we know its exp₹≈₩πected price at its maturity, we >✘must compute its present value at γt = 0. It means that we m$σust know the discount rate≈≥γ for the derivative.
Unfortunately, in the re®♥ε∞al world, we could hardly know any ofδ∞π↑ them. This is why the risk-neu↔©γtral valuation is very useful←≥, as described by Hull (2011): "In a world where investors are risk n÷¶♣eutral, the expected retur≠≤↕n on all investment as±sets is the risk-free rate of int¥≠<erest, r. The reason•÷ is that risk-neutral investors dε§ o not require a premium to induce t≤™hem to take risks. It is also true >↕≈that the present value of any ca≤®sh flow in a risk-ne≥≈utral world can be obtain₩'ed by discounting its ∑≈£expected value at the risk-free ratσ← e. The assumption that the ≈•♥$world is risk neutral does, therefo&¶γre, considerably simplify the analysi☆"§s of derivatives."
A derivative can be va≈★lued using risk-neutral valuation by✘♥∑ using the following pβ↕rocedure (Hull 2011)$δ∏:
1. Assume that the expected return fro ÷Ωm the underlying asset is t₩↓>∏he risk-free interest rate→α≥, r (i.e., assume μ = r).
2. Calculate the expected♥← payoff from the derivative.↓©
3. Discount the expected α≈ payoff at the risk-free←±≥ interest rate.
Hull (2011) provides±π some great comments on the r≤↕isk-neutral valuation: It is importa®♥≠αnt to appreciate tha←♠t risk-neutral valuation is merely§σ an artificial device for obtaining±↕ solutions to the BS d₽φ&•ifferential equation. T↓ ©he solutions that are o✘✘btained are valid in all w₽•≈ orlds, not just those where investo☆✔rs are risk neutral. When we move₹β from a risk neutral worldπ•↔ to a risk-averse world, ↕☆σσtwo things happen. Th§₽σσe expected growth rate in&¥♠☆ the stock price changes and ↔≠♣the discount rate that musλ♠±t be used for any payoffs from the derλβ ivative changes. It hγ ↕₩appens that these two changes alwaλ×Ωys offset each other exactly.
We will next derive th ♣₹÷e price of European call o&ε>¥ption, C, using the risk-neut✘→ral valuation.
6 Black-Scholes Formula for option®>β pricing
The expected value of a European '→ call option at maturity is E[max≠∏↔(S(T) – K, 0)], wher♥₹e S(T) is the stock price ✘ εat t, and K is the strike p↓≈®≈rice. With risk-neutral valu€ε' ation, we assume tha↕ t the expected return of'∞✔✘ S is the risk-free rate r,₩✘♦ and the discount rate for the σ₽call option is also r. At ma★∏turity (t = T), the price of thλ↔e call option is theref←σ →ore:
Since S follows a logn↕≤ ormal distribution, it is not →α→hard to compute E[max(S(T) –δ∞ K, 0)] and doing that gives the fa™ε↑mous Black-Scholes formula for option pricing. For call and put ✔β©δoptions, their prices$ C and P are:
With the help of a computer, we only need to provide values of the 5 in$♠puts: current stock price S(0), strΩ§ike price K, time to maturi¥>ty T, risk-free interest rate r,★÷♣ and the standard deviation of the stoc₽¥λk σ. By using these formulas, it wil∑≤¥l be very convenient t✔&♣o find the prices for both Euro₽δ€pean call and put options.
But we must understand the meaning beh♠β€φind these formulas. For any option, two ε₩βuncertainties must be considered when αλφpricing it:
1. Whether this option w₹φγill be in-the-money at maturity. For ₩÷ example, if I purchase ¶§a call option, but the Ω÷price of the underlying asset S ♣®∞is lower than the strike price K at T,α→☆ then this option has no valε∞ue.
2. If the option is in-the-mon₩↑★ey and we exercise it, hπ→&∞ow much our payoff will be. ∑δ≤ For example, with a call option, if ÷ the stock price is 110 φ¥>and the strike price isλ§α 100, then by exercising, we earn←→λ 10 per share; if the stock α↓¥price is 20, we earn ∑✘÷>20 per share.
These two uncertainties are related to×€< N(d_1) and N(d_2) in the ∑↓¶BS formula.
We explain them using th✔Ωα©e call option. N in the BS formu♥'✘"la is the cumulative density f✘←φεunction of the standard nor§±✔mal distribution. Therefore, N(d_1€∞) and N(d_2) represent tw ™©o probabilities. In particular, N(d_2) ≥ §is the probability that the option wiσ' φll be exercised in the risk-neπ ₽utral world, i.e., it is ≤←∏prob(S(T) > K). It follows that the¶←£∞ second term in C, i.e., Kγ♣e^(-rT)N(d_2) is the present val∞₹£₹ue of the expected cost of exerci∏δ" sing the option at m✔≥aturity.
As for N(d_1), it is less intuitive t≈→&han N(d_2). Let's put N(d_1) ↓∏>aside shortly and check the fi ÷✘rst term in C first. Since th☆₩σe second term of C i>←αφs the expected cost, the first☆♥γ term of C must be the expected payo×↕ff (discounted to presen←≠¶t) when the call option is exercis€§₽¥ed at T. Since we will onδ™ly exercise the option wh≤φen S(T) > K, then the expected v&↓alue of this option conditioning on e↓'xercise is E[S(T) | S(T) > K]. Multi>< plying this conditional $☆©expectation with the ≥★probability of exercise∑¥π N(d_2) and then discounting it to ∑↕→today will give the f€σirst term of C. Then we have
Substituting S(0) with€δ e^(-rT)E[S(T)] in the equ♦∞ation above derives
Since E[S(T) | S(T) > K] > E[S(T)], we •₽∑have N(d_1) > N(d_2). σ♠©We then interpret N(d_1) as thΩ₩e probability of exercise weigh£₩ted by the stock price &® in the risk-neutral world. This is§∏ because unlike the cost whi£ch is independent of S, tββhe expected payoff is a fun>♣ction of S.
There is indeed another, yet mβπ¥ore mathematical, interprε↓♠Ωetation about N(d_1). It is the proba¥♣bility of exercising the opt♣±σion in a risk-averse world where the ma✘π§rket price of risk i↔σβs measured by σ and th• βe stock is used as the numeraire.♦®✔ It is perfectly fine to skiε®±≤p this interpretation ε&as understanding it requires the® knowledge of change of measu↔$↓ re, equivalent martingale measure and σ≈↕☆change of numeraire. These conce>₹pts are beyond the scope of this arti₩¶→cle. Interested readers are recomme→★εnded to take a look at Joshi (2008) fασγor further discussio→ ×€n.
If we take the partial d✔φerivative of C against S,φ ÷ it turns out that N(d_1)±"β happens to equal ∂C/↔∑>∂S. In practice, some people₽ & therefore interpret N(d_1) as the &&®measure of how sensitive of C t>¥o the change of S, althε≈ough this interpretation is ↕≥±not quite correct. Perhaps you will ask: the B$σ™≤S pricing formula only gives a theore₩¥tical price based on various strict as→¶sumptions, is it useful £ in the real world? Can people re₹β∞ally build profitable strategies by •™ βutilizing the difference be™§→©tween the theoretical pric✔∏e and the actual price? The key value of the BS pricing for&φmula is that it provide∏s a analytical tool. It allow"→s us to calculate various risk♣ε<± exposures that the option >↕♦"experiences. This is very ↓"✔important for investλ♥πors who want to use options (as we♣≥♠ll as other derivatives) in their po >λrtfolios. With the BS formula (i.e., takin™∞¶g partial derivatives), it is♠β easy to calculate how derivative's pri ≈ce is impacted by the price of th β≠e underlying asset, time, interest rate↓β, and the volatility. Thi≠"s gives the exposures of the derivati☆₩÷ve to these risk factors:
They are also referred to Greeks sin<☆€φce they are all Greek letters (note V>©εega is not a Greek letter). We wil÷Ωl explain them in detail in♠γ' another article in the future✘£✘'. Another important usage of the &$≈"BS formula is to calculate the impliedσ∑₹ volatility of the underlying asβ→set. In the BS formula, all inpu≥₹←t parameters other t÷han σ are very clear to all the inδ♣♥↓vestors, but people γ€may take different values for σ. Byε using the option's price, it isε< possible to inversely deduce the v∞®alue of σ agreed by Ω¥most of the investors (as it is c≥δβalculated from the option's price) an ¶βΩd this is called the implied volatilit§✘y. The most famous example for imp←Ω ★lied volatility is the &♥★VIX. VIX is the ticker symbol fo♣βαπr the Chicago Board Options Excha π♦nge (CBOE) Volatility Ind÷×&"ex, which shows the mar"₽ ket's expectation of 30-day volatili₩→×ty. It is constructed using t≤≈δ♠he implied volatilities ♦§ of a wide range of S&∏↔♠P 500 index options.
7 Summary
This article starts with the genera εΩl form of Ito's lemma. The ≈₹∏®derivation shows that• a function of a Ito ≈≥process is also a Ito process, and the ©λrandomness of the twoλ≈ processes is the same and comes fro€✔¥m the same underlying Browni♦$☆↓an motion. This fact i×πε®s crucial to the derivation of the BS ☆$≥§difference equation.
With the help of Ito's lemma,♥≈£₩ it is easy to solve the geometric∏₩↕φ Brownian motion, which iγ✔s used to model stock pricγ←e movement. Under this assumptio↕≥÷n, stock price follows a l™•✔ ognormal distribution and↔₩∏ this is one of the assumptio→☆ ns used by the BS model.
Finally, by using delta hedginε♥≈g, we can construct a portfolio of th♣∏e underlying stock a≥<©'nd its derivative to eliminat<αe the uncertainty and thereforeβ ₹± derive the BS differential equ§₽&↓ation. With delta hedging, the risk φφ preference of investors, μ, is alsoσφ♠ removed from the BS equ¶≈ation. This leads to a very important$≤& method to price the opti →™on, the risk-neutral valuation. It i₹≥s applied to derive the prices for bot>≥h the European call and put opt♣™ ions and we also explain α∞ ✔how to interpret these formulas.±<✔ At the end of the article, we ∞↑₽βoffer some discussionεα& about how the BS formula is used in pr♠★σactice. It is used to measure the expoβλ∞γsures of the derivative to va↔$rious risk factors, and this can b€♦↔e of great value to investor≈♥s who utilize options as an investment ₩™Ωinstrument.
References
Hull, J. C (2011). Options, ✔$ Futures, and Other Derivatives >β(8th ed.). Prentice Hall.
Joshi, M. S. (2008). The Concepts €φand Practice of Mathematical ♥®♥Finance (Mathematics, Finance andε> Risk) (2nd ed.). Cambridge Univ✔♥★£ersity Press.
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